Classical Elastic Model Research Articles

Multi-body dynamics-based sensitivity analysis for a railway vehicle

A railway vehicle is a complicated mechatronic system. Its mechanical structure and connections are key influences on its performance. When a railway vehicle runs at very high speeds its dynamic performance becomes of considerable interest. Therefore, the identification of the design factors that significantly impact on the vehicle’s running behavior, such as safety, stability, comfort and reliability, is a key step towards vehicle design optimization. The optimal design of a railway vehicle is difficult to perform using existing methods due to the complexity inherent in rail/track interactions. Thus, performing a sensitivity analysis of a railway vehicle can be seen as a bridge between performance analysis and an optimal design. This paper presents a sensitivity model that can be used to analyze the performance parameters of a railway vehicle in terms of variation of its design parameters. First, the railway vehicle is described as a multi-body system that can be broken down into its component parts and then, the dynamic behavior of the system is investigated. To simplify the model, a spring/damper connection is used in the classic Hertz nonlinear elastic contact model to represent the force in the normal direction at the wheel/rail contact point. Finally, based on the adjoint variable method, the equations that constitute the sensitivity model of the railway vehicle are derived and analysis software is developed. A case study shows the reliability of the proposed approach.

Effect of Protein-Induced Spontaneous Curvature on Membrane Surface Tension

Adsorption of proteins onto membranes can alter the local membrane curvature. This phenomenon has been observed in biological processes such as endocytosis, tubulation and vesiculation. In this letter, we show that the classical elastic model of lipid membranes cannot account for simultaneous changes in shape and membrane tension due to protein adsorption in a local region, and a viscous-elastic formulation is necessary to fully describe the system. Using the viscous-elastic model, we show that protein adsorption can not only induce curvature but also alter membrane surface stress. Using the viscous-elastic model, we show that the lipids flow to accommodate the change in membrane curvature. Finally, at the end of protein adsorption process, the system has a residual stress to balance the difference between the actual mean curvature and imposed spontaneous curvature. Surface stress effects are local and change only in the protein patches, however, curvature changes may be non-local and remain significant for large separations between the protein patches.

Potential of a piezoelectric energy harvester from sea waves

A sea wave energy harvester from the longitudinal wave motion of water particles is developed. The harvester consisting of a cantilever substrate attached by piezoelectric patches and a proof mass is used to collect electrical energy owing to the electromechanical coupling effect of the piezoelectric patches from the longitudinal wave motion. To describe the energy harvesting process, a mathematical model is developed to calculate the output charge and voltage from the piezoelectric patches according to the Airy linear wave theory and classical elastic beam model. Results show that the mean value of the generated power increases with the increase in the ratio of the width to the thickness of the cantilever, the wave height, the sea depth (which equals to the cantilever height in this study), the ratio of the proof mass to the cantilever mass, and the ratio of the sea depth to the wave length. A value of the power up to 55 W can be realized for a practical sea wave with the values of the sea depth, wave height and wave length to be 3 m, 2 m, and 15 m, respectively. The collected power harvesting with respect to different categories of the sea waves are provided. Our simulations also show the generated electric power can be further increased by an increase in dimensions of the harvester considering the scale effect. This research develops a new technique for energy harvesting from sea waves by piezoelectric energy harvesters.

On the dilatational wave motion in anisotropic fractal solids

This paper reports a study of wave motion in a generally anisotropic fractal medium (i.e. with different fractal dimensions in different directions), whose constitutive response is represented by an isotropic Hooke's law. First, the governing elastodynamic laws are formulated on the basis of dimensional regularization. It is discovered that the satisfaction of the angular momentum equation precludes the implementation of the classical elasticity theory which results in symmetry of the Cauchy stress tensor. Nevertheless, the classical elastic constitutive model can still be applied to explore dilatational wave propagation; in such a case, the angular momentum balance is “trivially” satisfied. The resulting problem, of eigenvalue type, is solved analytically. A computational finite element method solver is also developed to simulate the problem in its one-dimensional (1d) 1D form, it is validated by the reference solutions generated through the modal analysis. A 3D finite-difference-based solver is also developed and the obtained results match those of the 1D simulation.

Existence of weak solutions in elasticity

Solvability and uniqueness of solutions to the problems of equilibrium, vibration and dynamics in a weak setup for classical and nonclassical models of linear elasticity are established in a unified framework sufficiently flexible to accommodate new elastic models.

Buckling analysis of shear deformable nanorods within the framework of nonlocal elasticity theory

This article provides useful insights into the ealstica type buckling phenomenon of shear deformable nanorods. In the analysis, a rod is considered to be a prismatic, inextensible column, whose constitutive equation corresponds to a differential type of the nonlocal elasticity. The strongly nonlinear governing equation of the buckling of the rod is established and solved by the homotopy perturbation method. The governing equations of buckling take into consideration the effects of small length scale and transverse shear deformation. The validity, convergence and accuracy of the solutions are partly established by comparing them with known classical elastic model solutions. The numerical results show that an increase in the small scale parameter, as well as the transverse shear deformation, gives rise to an increase in post-buckling deformation, and a decrease in the buckling load.

Transient dynamic analysis of a fluid-saturated porous gradient elastic column

The dynamic response of a fluid-saturated porous gradient elastic column to a transient disturbance is determined analytically and numerically. The basic dynamic theory of a fluid-saturated poroelastic medium due to Biot is modified by replacing the classical linear elastic model of the solid skeleton by the simple gradient elastic model of Mindlin with just one elastic constant (internal length scale) in addition to the classical ones. Thus, the new theory, which is presently restricted to the one-dimensional case, can take into account the microstructural effects of the solid skeleton. After the establishment of appropriate boundary and initial conditions, the one-dimensional dynamic column problem is solved analytically with the aid of the Laplace transform with respect to time. The time domain response is finally obtained by a numerical inversion of the transformed solution. The effect of the solid microstructure on the response is assessed and discussed.

Vibration of Single- and Double-Layered Graphene Sheets

Free vibration of single- and double-layered graphene sheets is investigated by employing nonlocal continuum theory and molecular dynamics simulations. Results show that the classical elastic model overestimated the resonant frequencies of the sheets by a percentage as high as 62%. The dependence of small-scale effects, sizes of sheets, boundary conditions, and number of layers on vibrational characteristic of single- and double-layered graphene sheets is studied. The resonant frequencies predicted by the nonlocal elastic plate theory are verified by the molecular dynamics simulations, and the nonlocal parameter is calibrated through the verification process. The simulation results reveal that the calibrated nonlocal parameter depends on boundary conditions and vibrational modes. The nonlocal plate model is found to be indispensable in vibration analysis of grapheme sheets with a length less than 8 nm on their sides.

Small scale effects on the stability of carbon nano-peapods under radial pressure

In this paper, a nonlocal elasticity formulation is presented for analyzing the instability of nano-peapods by modeling the nanotube as a shell. Using the nonlocal elasticity theory, the small scale characteristics of Carbon Nano-Peapods (CNPs) are taken into account. While the classical elastic shell model overestimates the critical pressure for the onset of structural instability of carbon nanotubes, the obtained results show that the nonlocal elastic shell model for nano-peapods can potentially provide better predictions. According to the results, it is concluded that the presence of C 60 inside (10,10) Carbon Nanotubes (CNTs) significantly increases the stability resistance of the single CNT.

Natural Frequency Computation Method of Nonlocal Elastic Beam

After adopting the constitutive equations of the nonlocal elastic media in the form of Eringen, and making use of the Laplace transformation, the vibration governing equation of nonlocal elastic beam in the Kelvin media are established. Unlike classical elastic models, the stress of a point in a nonlocal model is obtained as a weighted average of the field over the spatial domain, determined by a kernel function based on distance measures. The motion equation of nonlocal elastic beam is an integral differential equation, rather than the differential equation obtained with a classical local model. Solutions for natural frequencies and modes are obtained. Numerical examples demonstrate the efficiency of the proposed method for the beam with simple boundary conditions.

Mathematical analysis for the peridynamic nonlocal continuum theory

We develop a functional analytical framework for a linear peridynamic model of a spring network system in any space dimension. Various properties of the peridynamic operators are examined for general micromodulus functions. These properties are utilized to establish the well-posedness of both the stationary peridynamic model and the Cauchy problem of the time dependent peridynamic model. The connections to the classical elastic models are also provided.

Vibration and length-dependent flexural rigidity of protein microtubules using higher order shear deformation theory

Microtubules are hollow cylindrical filaments of the eukaryotic cytoskeleton characterized by extremely low shear modulus. A remarkable controversy has occurred in the literature, regarding the length dependence of flexural rigidity of microtubules predicted by the classical elastic beam model. In this study, a higher order shear deformable beam model for microtubules is employed to study unexplained length-dependent flexural rigidity and Young’s modulus of microtubules reported in the literature. The formulation allows for warping of the cross-section of the microtubule and eliminates the need for using arbitrary shear correction coefficients as in other theories. It is showed that vibration frequencies predicted by the present parabolic shear deformation theory (PSDT) are much lower than that given by the approximate isotropic beam model for shorter microtubules, although the two models give almost identical results for sufficiently long microtubules. It is confirmed that transverse shearing and the warping of the cross-section of microtubules are mainly responsible for the length-dependent flexural rigidity of an isolated microtubule reported in the literature, which cannot be explained by the widely used Euler–Bernoulli beam model. Indeed, the length-dependent flexural rigidity predicted by the present model is found to be in qualitative agreement with the existing experimental data ( Kurachi et al., 1995; Pampaloni et al., 2006). These results recommend that the parabolic shear deformation-beam theory offers a unified simple 1D model, which can capture the length dependence of flexural rigidity and be applied to various static and dynamic problems of microtubule mechanics.

Inelastic Interface Damage Modeling with Friction Effects: Application to Z-Pinning Reinforcement in Carbon Fiber Epoxy Matrix Laminates

This article presents the implementation of a new inelastic damage model able to carry out simulation of initiation and evolution of damage in the Z-pinned laminated composite structures with friction effects. The classical elastic damage model is modified to an inelastic model with friction effects obeying the simple Coulomb friction criterion. The main idea is the modification of strain energy parameter by introducing sliding and friction parameters. The simulations of single Z-fiber pull tests highlight the effectiveness of the proposed model for micro-scale predictions.

Parametrization of atomic force microscopy tip shape models for quantitative nanomechanical measurements

The uncertainty of the shape of the tip is a significant source of error in atomic force microscopy (AFM) based quantitative nanomechanical measurements. Using transmission electron microscopy, scanning electron microscopy, or tip reconstruction images, it is possible to parametrize the models of real AFM tips, which can be used in quantitative nanomechanical measurements. These measurements use algorithms described in this article that extend classical elastic, plastic, and adhesive models of contact mechanics. Algorithms are applicable to the tips of arbitrary axisymmetric shapes. Several models of AFM tip have been utilized. The goal of tip model parameterization is to develop AFM tip-independent quantitative mechanical measurements at the nanometer scale. Experimental results demonstrate independence of the AFM measurements from tips and their closeness to bulk measurements where available. In this article the authors show the correspondence between microtensile, nanoindentation, and AFM based indentation for measuring the modulus of a spin on low-k dielectric polymer thin film. These results provide a confidence that with the described procedures AFM can be used for reliable quantitative nanomechanical measurements.

Supercoiling and Denaturation of DNA Loops

We study the statistical mechanics of small DNA loops emphasizing the competition between elasticity, supercoiling, and denaturation. Motivated by recent experiments and atomistic molecular dynamics simulation, we propose a new coarse-grained phenomenological model of DNA. We extend the classical elastic rod models to include the possibility of denaturation and nonlinear twist elasticity. Using this coarse-grained model, we obtain a phase diagram in terms of fractional overtwist and loop size that can be used to rationalize a number of experimental results which have also been confirmed by atomistic simulations.

AXI-SYMMETRIC WAVE PROPAGATION OF CARBON NANOTUBES WITH NON-LOCAL ELASTIC SHELL MODEL

A non-local elastic shell model is proposed for the first time in this study for considering the small-scale effect in axi-symmetric wave propagation in carbon nanotubes (CNTs). Two coupled radial and longitudinal modes, and one decoupled torsional mode, are derived from the developed non-local shell model. The small-scale effect on wave propagation is numerically studied and discussed. In addition, the cut-off frequency based on the non-local shell model is obtained, which is found to be free of the small-scale effect. It is interesting to note that only one asymptotic phase velocity exists in the CNTs using the non-local shell model, whereas two asymptotic phase velocities can be predicted using the classical or local elastic shell model. It is hoped the research presented herein can be used as a benchmark for future studies on the wave propagation of CNTs with non-local continuum models.

DNA looping by two-site restriction endonucleases: heterogeneous probability distributions for loop size and unbinding force

Proteins interacting at multiple sites on DNA via looping play an important role in many fundamental biochemical processes. Restriction endonucleases that must bind at two recognition sites for efficient activity are a useful model system for studying such interactions. Here we used single DNA manipulation to study sixteen known or suspected two-site endonucleases. In eleven cases (BpmI, BsgI, BspMI, Cfr10I, Eco57I, EcoRII, FokI, HpaII, NarI, Sau3AI and SgrAI) we found that substitution of Ca2+ for Mg2+ blocked cleavage and enabled us to observe stable DNA looping. Forced disruption of these loops allowed us to measure the frequency of looping and probability distributions for loop size and unbinding force for each enzyme. In four cases we observed bimodal unbinding force distributions, indicating conformational heterogeneity and/or complex binding energy landscapes. Measured unlooping events ranged in size from 7 to 7500 bp and the most probable size ranged from less than 75 bp to nearly 500 bp, depending on the enzyme. In most cases the size distributions were in much closer agreement with theoretical models that postulate sharp DNA kinking than with classical models of DNA elasticity. Our findings indicate that DNA looping is highly variable depending on the specific protein and does not depend solely on the mechanical properties of DNA.

Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum models

Nonlocal elastic beam and shell models are developed and applied to investigate the small scale effect on buckling analysis of carbon nanotubes (CNTs) under compression. General and explicit solutions are derived and expressed in terms of the solutions via local or classical elastic models, in which the scale effect is not accounted, to reveal the small scale effect on CNTs buckling results. The dependence of the scale effect with respect to the length, radius, and buckling modes of CNTs is clearly established and observed from the universal solutions derived in the manuscript. It is clearly seen from the results that the buckling solutions for CNTs via local continuum mechanics are overestimated and hence the scale effect is indispensable in providing more accurate results for mechanical behaviors of CNTs via continuum mechanics.

Effect of tetragonal distortion introduced by anti-site defect configuration on additional hardening in off-stoichiometric Al-rich Ni 3Al alloys

A theoretical approach to the hardening mechanism of L1 2–Ni 3Al compound with Al-rich off-stoichiometric composition is attempted based on the classical elastic interaction model between an edge dislocation and a unit cell with an equi-atomic composition embedded in Ni 3Al. A tetragonal distortion is expected to arise from a local anisotropic atomic configuration in the unit cell which is recognized as a face-centered tetragonal-L1 0 or distorted B2 NiAl structure around an anti-site Al atom. According to the study by Cochardt et al. (Acta Metall 1955;3:533), it is found that the maximum point force F m on an edge dislocation from the local L1 0 unit cell increases with decreasing the c i / a i ratio of the L1 0 structure toward the stable B2 structure, which is expected from the authors’ recent electronic structure calculations on the stability of L1 0–NiAl relative to B2-NiAl. This increment of F m by decreasing the c i / a i ratio accounts for a larger hardening in Al-rich Ni 3Al than that in Ni-rich Ni 3Al at 77 K.

Numerical Procedure for Elasto-Plastic No-Tension Model

In the present work a generalization of the classical elastic no-tension model accounting for irreversible crushing strains is proposed. The model considers limited tensile and compressive strength; in particular, fracture strain arises when the tensile strength is reached, while plastic strain can be developed because of the limited compressive strength. As a consequence, inelastic strains are partitioned into the sum of a positive semi-definite fracture strain and a plastic (crushing) strain, which are both assumed to fulfill a normality rule. A numerical procedure, based on the backward Euler time-integration scheme, is proposed; the time step is solved adopting a strain driven predictor-corrector scheme. The consistent no-tension elastoplastic tangent is computed. Then, the model is implemented in a finite element code. Numerical applications, concerning the study of the behavior of some significative masonry structures, are presented.